On the capacity of isolated, curbside bus stopsweihuagu/curbside-bus-stop.pdf · It will be assumed that bus stops operate in the steady-state, such that the arrival process and the

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On the capacity of isolated, curbside bus stops

Weihua Gu *, Yuwei Li, Michael J. Cassidy, Julia B. Griswold

416 Mclaughlin Hall, Department of Civil and Environmental Engineering, University of California,

Berkeley, CA 94720, United States

Abstract

The maximal rates that buses can discharge from bus stops are examined. Models were developed

to estimate these capacities for curbside stops that are isolated from the effects of traffic signals.

The models account for key features of the stops, including their target service levels assigned to

them by a transit agency. Among other things, the models predict that adding bus berths to a stop

can sometimes return disproportionally high gains in capacity. This and other of our findings are

at odds with information furnished in professional handbooks.

Keywords: bus stop capacity; bus stop queueing

1 Introduction

While serving passengers at a busy stop, buses can interact in ways that limit their discharge

flows. This can degrade the bus system’s overall service quality (Fernandez, 2010; Fernandez and

Planzer, 2002; Gibson et al., 1989).

The present paper explores the bus discharge flows that can be achieved at stops where buses

dwell curbside to load and unload passengers. We will examine stops that are isolated from the

influences of traffic signals and other bus stops; where sufficient space exists for storing the bus

queues that can form immediately upstream of the stops; where bus movements in and around the

stops are not affected by other (e.g. car) traffic; and where bus overtaking maneuvers are

prohibited, both within any bus queues immediately upstream, and at the stops themselves, should

multiple berths (i.e. bus loading areas) exist there.1

The rates that buses can discharge from stops of this kind depend in part on the target service

level chosen by the transit agency. In this paper, we use a metric of service level called the failure

rate, , defined as the probability that a bus arriving to a stop is temporarily blocked from using

it by another bus. Though other service metrics (e.g. average bus wait time) are possible, is

the metric featured in professional handbooks (e.g. TRB 2000 and 2003). Intuitively, the bus

* Corresponding author. Tel.: +1 (510) 931-6646; fax: +1 (510) 643-8919.

E-mail address: [email protected]. 1 Cities often enact this prohibition because an overtaking bus can disrupt car traffic in the adjacent lane(s).

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discharge flow increases as increases, and is highest when a bus queue is always present at the

stop’s entrance, i.e. when .2

In light of this influence, we shall define bus-stop capacity as the maximal rate that buses can

discharge from a stop for a specified threshold of . This definition is common in the literature

(see again TRB 2000 and 2003). Our findings, on the other hand, are largely at odds with earlier

publications, as we shall see. We shall arrive at these findings by developing (and evaluating)

models that predict bus-stop capacities as functions of not only , but also bus arrival process

and bus service time distribution.

A review of earlier work is furnished in the following section. Present findings in regard to stops

with only one berth are provided in Section 3. Findings on multi-berth stops are in Section 4.

Practical implications are discussed in Section 5.

2 Literature Review

The Highway Capacity Manual (TRB, 2000) reports that the capacity of a single-berth stop is

inversely proportional to the sum of i) the bus’ average service time; and ii) a second term that

accounts for both the variation in this service time and the .3 With this latter term, a stop’s

capacity increases with increasing , but only to a point. Curiously, the formula in the Highway

Capacity Manual (henceforth HCM) predicts that capacity is maximal when reaches 0.5.

Intuition, on the other hand, tells us that single-berth capacity is maximal when a bus queue

always persists upstream; i.e., when is 1. Of further concern, the current edition of the HCM

omits any discussion on the influence of the bus arrival process on stop capacity.4

For a multi-berth stop, the HCM takes capacity to be the product of the single-berth capacity and

the number of “effective” berths. The HCM furnishes values for this latter term that result in

steadily diminishing returns to capacity, meaning that each new berth that is added to a stop will

return less than a proportional increase in the stop’s capacity (see Table 27-12 of the TRB 2000).

Presumably, this is to account for the disruptive bus interactions that can occur at multi-berth

stops (see our discussion of the “blocking effect” in Section 4.1). However, the inefficiencies

brought with each added berth are assumed in the HCM to be independent of all other factors,

including: , bus arrival process, and service time variation.

Much of the above is at odds with our present findings (see Sections 3 and 4). What thus appear

to be shortcomings of the HCM take on greater significance because they are repeated in the

Transit Capacity & Quality of Service Manual (TRB, 2003). This latter handbook will reportedly

2 If buses were controlled so that their arrival headways and service time at a stop were perfectly

coordinated, the stop could, in theory, always be occupied without queues forming. The in this

idealized (and unrealistic) case would therefore be zero, though the bus discharge flow would be high. 3 The second term involves both: the one-tail standard normal variate corresponding to ; and the

coefficient of variation of bus service time (see Equation 27-5 of TRB, 2000). 4 Although an earlier edition of the HCM includes a multiplicative adjustment factor that reportedly

accounts for variations in bus arrival headway, the factor seems instead to account for (see Equation 12-

7 and Table 12-17 of TRB, 1985).

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supplant discussion of transit systems in future editions of the HCM. The same ideas, moreover,

have found their way into the Transportation Planning Handbook (ITE, 1999).

Critiques of these capacity formulas already appear in the literature. Gibson et al. (1989), for

example, argues that the complex stochastic processes at real bus stops limit the usefulness of

HCM formulas. Fernandez and Planzer (2002) reports that the formulas tend to under-predict

field-measured estimates of stop capacity. These findings are useful in that they highlight certain

influences on bus-stop capacity. Yet, they do little to quantify these influences.

Similarly, studies to increase the capacity of a multi-berth stop by either dispatching buses in

certain ways (Gardner et al., 1991; Szász et al., 1978), or by reconfiguring the stop’s geometry

(Gibson et al., 1989; St. Jacques and Levinson, 1997; etc.) offer only limited insights into cause

and effect. The same is true of past efforts to estimate the parameter values for describing bus

arrival processes (Danas, 1980; Fernandez, 2001; Ge, 2006; Kohler, 1991) and service time

distributions (Ge, 2006; St. Jacques and Levinson, 1997).

3 Single-Berth Stops

It will be assumed that bus stops operate in the steady-state, such that the arrival process and the

service time distribution are both time-invariant, and that the long-run average bus arrival rate

never exceeds the stop’s capacity when is 1. In this steady-state, the average bus inflow to the

stop always equals the average outflow.

Although some empirical studies show that bus arrivals at stops follow a Poisson process (Danas,

1980; Ge, 2006; Kohler, 1991), other studies (e.g. Fernandez, 2001) argue that this is not always

the case. To simplify our analysis and highlight the findings, we start by assuming two special

cases in regard to the bus arrival process: Poisson arrivals (in Section 3.1), as can occur when the

stop serves multiple bus routes; and uniform bus arrivals (in Section 3.2), as may occur, at least in

theory, when the stop serves a single route with buses that are rigidly controlled. Finally, Section

3.3 examines the case of a more general bus arrival pattern. Capacity formulas will be furnished

for each of these three cases.

3.1 Poisson Bus Arrivals

In the steady-state, Poisson bus arrivals to a stop satisfy the PASTA (Poisson Arrivals See Time

Averages) property; see Wolff (1982). This implies that is equal to the fraction of time that

the stop’s single berth is utilized. This utilization fraction is the ratio of bus inflow, , to the

single-berth stop’s maximal service rate (i.e. the inverse of the average time that each bus spends

serving passenger boarding and alighting movements). We denote this maximal service rate as .

Thus, for ,

. (1)

Since can be viewed equivalently as the stop’s capacity for a specified ; and since is the

stop’s output flow when ; the ratio will henceforth be termed the normalized capacity.

4

As per intuition, (1) shows that single-berth capacity is maximal when . It further shows

that for Poisson bus arrivals, capacity is independent of the variation in bus service time (for

boarding and alighting movements). This independence turns out not to hold in general, however,

as we shall see next.

3.2 Uniform Bus Arrivals

Assume now that the bus arrival headways are deterministic and equal. Further assume that bus

service time follows an Erlang- distribution, which is a more general distribution than the

commonly-used exponential distribution (and has been observed in Ge, 2006 to be suitable at

some stops.) For this present case, our model does not have a closed-form solution. An analytical

model that can be solved numerically is derived in Appendix A. A simple, closed-form

approximation to the solution of this model is found to be:

, (2)

where is the coefficient of variation in bus service time.

Equation (2) came by fitting a curve to our numerical solutions over the range of ,

since this is consistent with the range of observed in the literature (St. Jacques and Levinson,

1997). The result satisfies intuitive boundary conditions for the relation between and .5

The inclusion of in (2) is logical, since for Erlang distributions, and this shows how

stop capacity for the case of uniform bus arrivals depends on the coefficient of variation in bus

service time as well as on .

To explore matters more deeply, relations generated from (2) are shown with solid curves in

Figure 1 for 0.1, 0.5, and 1. These curves collectively reveal that, for uniform arrivals and

for , capacity increases as the coefficient of variation in bus service time diminishes.

The curves further show that the maximal capacity of the stop (when ) is the same for all

. The case of corresponds to the perfect coordination of bus arrivals and bus service

time, as previously discussed in Footnote 2, such that . The curve in this idealized case

therefore reduces to a point, also as shown in Figure 1.

The relation for Poisson bus arrivals revealed in (1) is shown in Figure 1 as well; see the dashed

line. Comparing this dashed line against the solid curves reveals that for , capacity

also increases with diminishing variation in bus headways. (We can see this because the

coefficient of variation is 0 and 1 for uniform and Poisson bus arrivals, respectively).

5 These conditions are: i) if and ; and ii) if and if .

5

Figure 1 – Normalized capacity versus for single-berth stops;

comparisons between Poisson and uniform bus arrivals

3.3 General Bus Arrivals

We continue to model bus service time as above, and now use the Erlang- distribution to

describe a more general distribution for bus headways. A numerical solution was derived in

similar fashion to the uniform bus-arrival case, for which an approximation is found to be:

, (3)

where is the coefficient of variation of bus arrival headways.

From (3) we see that stop capacity is influenced by service time and headway variations. Readers

can verify that reductions in the coefficient of variation for either of these factors will increase a

stop’s capacity when , and ; e.g. one can fix either or and

obtain curves that are qualitatively similar (in their shapes and their relative positions) to the solid

curves in Figure 1.6

4 Multi-Berth Stops

Two competing effects, which we term the “blocking” and the “berth pooling” effects, are found

to influence the capacity of multi-berth stops, as explained in Section 4.1. The returns to capacity

from added berths are studied for two limiting cases that isolate the above effects and for a third,

more general case, all in Section 4.2. Further findings come by examining how returns to capacity

are influenced by coefficients of variation in bus service time and bus headway, as shown in

Section 4.3. For all these analyses, we will assume that the distribution of an individual bus’

service time (to load and unload passengers) is independent of the stop’s number of berths.

6 In addition to satisfying the conditions in Footnote 5, Equation (3) reduces to (1) for the case of Poisson

bus arrivals where . As an aside, analysis shows that (3) produces significantly lower capacities as

compared with the formulas of the HCM (TRB, 2000).

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Uniform bus arrivals, CS = 0.1

FRPoisson bus arrivals

Uniform bus arrivals, CS = 0.5

Uniform bus arrivals, CS = 1

λ/µUniform bus arrivals, CS = 0 (i.e., perfect coordination)

6

4.1 Two Competing Effects

Discussion begins with the blocking effect. A bus can enter a stop only when its upstream-most

berth is open. (At this time, the entering bus proceeds as far as possible until encountering the end

of the stop or a dwelling bus; and the entering bus will then dwell at the downstream-most

available berth for its entire time in the stop.) Similarly, a bus can discharge from a stop only after

all buses that were previously dwelling at that stop’s downstream berths have departed. This

blocking effect for entering and exiting a stop tends to diminish the stop’s returns to capacity

brought by added berths. The effect diminishes, however, when the load rate, , approaches 0,

where and is the number of berths at the stop.

We illustrate the second effect, berth pooling, with the following example. Consider two

independent, single-berth stops, each with equal bus arrival rate, , as shown on the left side of

Figure 2. (Dashed boxes in this figure denote berths, and shaded rectangles denote buses). If we

ignore the blocking effect, the fluctuations in bus arrivals would be better served by pooling the

two berths into a single, double-berth stop, as shown on the right side of Figure 2. Thus for the

same total bus arrival rate ( for both the left and right sides in the figure), this berth pooling

effect means that the double-berth stop would enjoy a lower than would the two single-berth

stops; i.e., the double-berth stop would have a higher capacity for a given . Berth pooling tends

to improve the stop’s returns to capacity brought by added berths. The effect diminishes, however,

when approaches its maximum, meaning when the input flow, , approaches the stop’s

maximal capacity (see Equation 4).

Figure 2 – Berth pooling effect

The above effects are countervailing: as approaches 0 or its maximum, one effect diminishes

while the other dominates.7 We will therefore isolate the two effects by examining multi-berth

stops under the two limiting cases for .

4.2 Returns to Capacity

We next explore the returns to capacity i) when is maximal; ii) when ; and iii) for the

general case when falls between these limits.

7 As per Footnote 2, an exception can occur under perfect coordination; i.e., when platoons of buses

arrive at uniform intervals and the service time is constant. In this case, neither blocking nor berth pooling

take effect and the is always zero.

bus waiting for entry

passenger platform

passenger platform

berth

passenger platform

λ

λ

λ

λ

empty stop

dwelling bus

7

4.2.1 Limiting case when is maximal

In this case, queued buses enter a stop in platoons of size , and the time required to serve a

platoon is the maximal bus service time across the platoon. The stop’s maximal capacity, , is

therefore:

, (4)

where is the expected value of the platoon service time; and is the cumulative

distribution function of the individual bus service time. The derivation of (4) is furnished in

Appendix B. Intuitively, the bus arrival pattern (to the rear of the queue) does not influence

capacity in this limiting case.

The average capacity per berth, , decreases with added berths, since increases

with . Thus from the first equality in (4), we see how the blocking effect can create decreasing

returns to capacity.

4.2.2 Limiting Case of

Computer simulation is used next to explore stop capacity under this second limiting case. The

logic of our simulation model is described in Appendix C. For the analysis to follow, bus service

time is assumed to follow the gamma distribution (a generalization of the Erlang distribution)

with , as recommended by St. Jacques and Levinson (1997). Bus arrivals are assumed to

follow a Poisson process, as if the stop were used by multiple bus lines. Simulations of other bus

arrival patterns and service time distributions yield qualitatively similar results.

The curves in Figure 3 display the normalized incremental change in stop capacity achieved for

each added berth, , for the first through the sixth berth. These curves are shown for near-

zero values of , since it is the assumed metric of interest and is a reasonable proxy for . (Note

that approaches zero when does so, and that the maximal value of one coincides with the

maximal value of the other.) The curves reveal that increases with each additional berth; i.e.,

that added berths bring increasing returns to capacity.

Figure 3 – Increasing returns to capacity caused by berth pooling effect

0

0.01

0.02

0.03

0.04

0.05

0 1 2 3

5th berth

4th berth

3rd berth

2nd berth1st berth

6th berth

FR (×10-5)

Δλ /μ

8

Although and might seldom approach zero in an urban setting, the finding calls into

question what handbooks have to say on the subject; i.e. the implication that added berths bring

decreasing returns to capacity does not hold in general. More interesting evidence in this regard

comes next.

4.2.3 General Case with Intermediate Values of

We now use our simulation model (see again Appendix C) to explore bus-stop capacities when

is between 0 and its maximum. Once again, bus arrivals are assumed to be Poisson, and service

time gamma-distributed with .

The curves in Figure 4a display the for the first through the sixth berth. These too are

shown as functions of , our chosen service metric and proxy for . The curves reveal how the

countervailing effects of blocking and berth pooling produce mixed results in terms of the

capacities returned by adding berths to a stop.

When is small (but not approaching zero), additional berths can produce increasing returns to

capacity, thanks to the berth pooling effect. For example, the figure shows that when ,

adding a second berth brings increasing returns. (Note that when , the curve for the

second berth lies above that for the first.) This favorable trend does not continue, however. Note,

for example, now the curve for the third berth lies below that for the second when .

Toward the other extreme (e.g. when ), the curves reveal that added berths produce

diminishing returns to capacity. This is because the blocking effect tends to dominate.

These findings are logical in light of what was unveiled for the two limiting cases. Yet, our

finding that returns to capacity vary with or runs counter to the HCM’s suggestion in this

regard; i.e. using a single set of numbers for “effective berths” evidently does not suffice for all

operating environments.

(a) Normalized incremental change in capacity versus

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Δλ/µ

FR

1st berth

6th berth5th berth4th berth3rd berth

2nd berth

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(b) Normalized capacity versus

Figure 4 – Normalized stop capacity and incremental change in capacity versus for

multi-berth stops with Poisson bus arrivals and gamma-distributed service time ( )

A graph like Figure 4a can be used in a number of practical pursuits. The same is true for variants,

like the curves of versus normalized capacity ( ) shown in Figure 4b. More will be said on

these matters in Section 5.

4.3 Variations in Service Time and Headway

Having explored the influences of and , we now examine how the returns to capacity are

influenced by the coefficients of variation in bus service time and bus headway. Simulation is

again used to this end.

4.3.1 Bus Service Time

We continue to assume that bus arrivals are Poisson and that service time is gamma-distributed.

Now, however, capacities will be explored for the range of

Figure 5a displays effects of on the for the first through the sixth berth when .

Note from the figure that increased returns to capacity come by adding a second berth to a stop

(i.e., the curve for the second berth lies above that for the first). This is again thanks to the

pooling effect at low . Further note that the curves for the second through the sixth berth

exhibit downward slopes. This reveals an inverse influence of on the returns to capacity.

Additionally, the downward sloping curves for in Figure 5b reveal how exerts an

inverse influence on stop capacity itself. These inverse influences become more dramatic as

increases. To illustrate, the above analysis is repeated, but for . Results are displayed in

Figures 6a and 6b.

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1

λ/µ

FR

1 berth

6 berths

5 berths

4 berths

3 berths

2 berths

10

(a) Normalized incremental change in capacity versus (b) Normalized capacity versus

Figure 5 – Normalized capacity and incremental change in capacity

versus for multi-berth stops with

(a) Normalized incremental change in capacity versus (b) Normalized capacity versus

Figure 6 – Normalized capacity and incremental change in capacity

versus for multi-berth stops with

4.3.2 Bus Headway

To explore how variations in bus arrival headway affect things, we will assume that: ;

bus service time is gamma-distributed with ; and bus headway is also gamma-distributed

with a coefficient of variation, , ranging from 0 to 1.

The curves in Figure 7a show that the first berth is relatively sensitive to ; i.e., when , the

diminishes precipitously with increasing . As a result, the for the second through

even the sixth berth is greater than that achieved by the first berth when is sufficiently high.

For example, we see that adding a second berth to a stop produces increasing returns to capacity

once comfortably exceeds 0.6. Once again, however, we find that a stop’s capacity for any

0.10

0.15

0.20

0.25

0.30

0.35

0 0.2 0.4 0.6 0.8 1

6th berth

5th berth

4th berth

3rd berth

2nd berth

1st berth

Δλ/µ

CS

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

6 berths

5 berths

4 berths

3 berths

2 berths

1 berth

λ/µ

CS

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 0.2 0.4 0.6 0.8 1

Δλ/µ

CS

6th berth

5th berth

4th berth

3rd berth

2nd berth

1st berth

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 0.2 0.4 0.6 0.8 1

λ/µ

CS

6 berths

5 berths

4 berths

3 berths

2 berths

1 berth

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diminishes as grows large; see Figure 7b. The above influences are found to disappear as

approaches 1.

(a) Normalized incremental change in capacity versus (b) Normalized capacity versus

Figure 7 – Normalized capacity and incremental change in capacity

versus for multi-berth stops with

5 Conclusions

The models presented in this paper account for key influences on the capacities of isolated,

curbside bus stops. They do so in ways that are more complete than what has been offered by

formulas in well-known handbooks. Through this more complete accounting come insights. The

insights have practical implications.

For example, the models predict that variations in bus service time tend to diminish stop capacity,

both for single- and multi-berth stops. (See Figures 1, 5b and 6b, and recall that an exception to

this occurs when buses arrive at a single-berth stop as a Poisson process.) This finding speaks to

the value of reducing service-time variations via the improved management of passenger

boarding and alighting. Means of doing this might include the use of wider bus doors, improved

loading platforms and off-board fare collection. Of course, these measures could also help reduce

the average service time, and this too would favorably affect bus-stop capacity.

In contrast to formulas in professional handbooks, the present models also account for the effects

of the bus arrival process at a stop. They predict that variations in bus headway can diminish stop

capacity (Figure 7b), but can in some instances favorably affect the returns to capacity brought by

a second through even a sixth berth relative to the returns from a single berth (Figure 7a). When

the variation in headway is high and the is low, adding a second berth to a single-berth stop

can bring increasing returns to capacity (Figures 4a and 7a). Knowledge of these cause and effect

relations can be useful when choosing the number of berths to be deployed at a curbside stop.

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1

Δλ/µ

CH6th berth5th berth

4th berth

3rd berth

2nd berth

1st berth

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

λ/µ

CH

6 berths

5 berths

4 berths

3 berths

2 berths

1 berth

12

To further illustrate the practical utility of our models, we ask the reader to refer again to Figure

4b. It displays relations between and normalized capacity for stops that range in size from 1 to

6 berths. Note how the curves in this figure can be used to determine the number of berths needed

to achieve targets for and stop capacity. Or, they can be used to estimate given bus arrival

rate and a specified number of berths. The figure can also help determine when it can be

advantageous to split a single stop with many berths into multiple adjacent stops. For example,

the reader can use Figure 4b to verify that, for a , splitting a 4-berth stop into two 2-

berth stops could increase capacity by nearly 15%. (That capacity is increased by splitting the

stop is clearly evident in Figure 4a, since at , the for the third and fourth berths are

lower than for the first and second berths.) Admittedly, this prediction assumes certain

idealizations; e.g. that both the bus arrival processes and the service time distributions are

comparable across the 2-berth stops; and that buses bound for one of these stops do not impede

buses bound for the other.

To be sure, all of our present models are idealized, particularly since they apply to isolated stops

operating in steady state. Yet in our view, these models represent a step toward better

understanding bus-stop operation. Work is ongoing in regard to stops: that are not isolated, but

are instead affected by traffic signals and other bus stops; that have limited space for storing bus

queues; and that allow bus overtaking. In the mean time, one may still use our models to develop

graphs that are similar to those shown here, but that are tailored to local conditions for target ,

variations in service time and headway, and so on.

Acknowledgement

Funding for this work was provided by the University of California Transportation Center and the

Volvo Research and Educational Foundations.

Appendix A

Analytical Solution to a Single-Berth Stop with Uniform Bus Arrivals and Erlang-

Service Time (in Section 3.2)

Here we furnish a solution by applying a more general result given by Gross, et al. (2008) for a

queueing system with generalized-Erlang distributed headways and service time ( ,

where and are the distributions of bus headway and bus service time, respectively)8.

This general result is:

, (A.1)

8 A generalized Erlang distribution is the convolution of independent but not necessarily identical

exponential random variables. Here a bus headway can be expressed as the sum of exponential

components that are independent but may not be identical; and a bus service time can be expressed as the

sum of such components.

13

where is the Laplace-Stieltjes transform of the cumulative distribution function (CDF) of

bus waiting time; is the rate of the -th exponential component of the

distribution; and is the -th complex root with negative real parts of the

following equation with argument :

. (A.2)

Equation (A.2) is also given in Gross, et al. (2008). The is the rate of the -th

exponential components of the distribution.

Since the means of the headway and the service time are

and

, respectively, we

set , so that the bus headway and the service

time are Erlang- and Erlang- distributed, respectively; and so that the bus arrival rate and the

service rate are and , respectively. Given that when approaches infinity, the limit of the

Erlang- distribution is a deterministic value, we let , so that the headway becomes constant.

Then (A.2) becomes

.

Let , such that the solution of the above equation is:

ambert

, (A.3)

where function ambert is the inverse function of , which is multi-valued in

the field of complex numbers, and has no closed-form expression; and is the imaginary unit.

By picking up the roots of ’s with negative real parts, plugging them into (A.1), and then

taking a partial-fraction expansion, we obtain:

, (A.4)

where are constant coefficients to be determined by:

. (A.5)

By applying the inverse Laplace transform on (A.4), we obtain the CDF of the bus waiting time:

.

Therefore the failure rate becomes . (A.6)

For any given , the last term of (A.6) is a function of . Thus we find the

relation between and . The results can be obtained numerically.

Appendix B

Derivation of Equation (4) in Section 4.2.1

For a fixed number of berths, , let be the platoon service time,

where is the service time of the -th bus in the platoon. All ’s are independent, identically

14

distributed random variables subject to the CDF of . Let be the CDF of . Thus

we have:

.

From the identity

, we have:

.

Appendix C

Simulation Algorithm for the Multi-Berth Stops Analyzed in Sections 4.2 and 4.3

First we introduce the following notation used in our simulation model:

– Headway (in minutes) between the arrivals of us and us , and is the system idle

time before the first bus arrives;

– Service time (in minutes) of us , not including the time that us waits to depart the stop

after it has finished serving passengers;

– The position (number) of the berth where us dwells to serve passengers; where berths

are numbered from the downstream to the upstream berth;

– Waiting time in the queue (in minutes) of us before it enters the stop;

– Waiting time in the berth (in minutes) of us after its service is finished; and

– Indicator that takes 1 if us fails to enter the berth immediately upon its arrival to the stop,

and 0 otherwise.

The dynamic equations describing our simulation model are:

For each

if

otherwise

if

otherwise

if

if

15

The and are inputs to the simulation. We assume that follows a gamma

distribution with mean and coefficient of variation . (For Poisson bus arrivals, and

is exponentially distributed.) We further assume that follows another gamma distribution

with mean and coefficient of variation . The simulation starts from an initial state in

which the stop is empty (i.e., and ) and ends at the same state to

diminish stochastic error. The resulting performance measure is obtained by averaging the .

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